What is the limit as n approaches infinity of n^{-2}e^{3n} ?

The limit as n approaches infinity of n^{-2}e^{3n} is infinity

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limit as n approaches infinity of n^{-2}e^{3n}

n \to \infty lim (n^{- 2} e^{3 n})

\infty

Solution steps

n \to \infty lim (n^{- 2} e^{3 n})

Simplify

n^{- 2} e^{3 n} : \frac{e ^{3 n}}{n ^{2}}

= n \to \infty lim (\frac{e ^{3 n}}{n ^{2}})

Apply L'Hopital's Rule

= n \to \infty lim (\frac{e ^{3 n} \cdot 3}{2 n})

Apply L'Hopital's Rule

= n \to \infty lim (\frac{9 e ^{3 n}}{2})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= \frac{9}{2} \cdot n \to \infty lim (e^{3 n})

Apply the Limit Chain Rule:

\infty

= \frac{9}{2} \cdot \infty

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

What is the limit as n approaches infinity of n^{-2}e^{3n} ?
The limit as n approaches infinity of n^{-2}e^{3n} is infinity